Whitham systems and deformations

We consider the deformations of Whitham systems including the "dispersion terms" and having the form of Dubrovin-Zhang deformations of Frobenius manifolds. The procedure is connected with B.A. Dubrovin problem of deformations of Frobenius manifolds corresponding to the Whitham systems of integrable hierarchies. Under some non-degeneracy requirements we suggest a general scheme of the deformation of the hyperbolic Whitham systems using the initial non-linear system. The general form of the deformed Whitham system coincides with the form of the "low-dispersion" asymptotic expansions used by B.A. Dubrovin and Y. Zhang in the theory of deformations of Frobenius manifolds.Comment: 27 pages, Late

Asymptotic Solutions of the Wave Equation with Degenerate Velocity and with Right-Hand Side Localized in Space and Time

We study the Cauchy problem for the inhomogeneous two-dimensional wave equation with variable coefficients and zero initial data. The righthand side is assumed to be localized in space and time. The equation is considered in a domain with a boundary (shore). The velocity is assumed to vanish on the shore as a square root of the distance to the shore, that is, the wave equation has a singularity on the curve. This curve determines the boundary of the domain where the problem is studied. The main result of the paper is efficient asymptotic formulas for the solution of this problem, including the neighborhood of the shore.Вивчається задача Кошi для неоднорiдного двовимiрного хвильового рiвняння зi змiнними коефiцiєнтами та нульовими початковими даними. Вважається, що права частина локалiзована в просторi та часi. Рiвняння розглядається в областi з межею (берегом). Вважається, що швидкiсть на березi зникає як квадратний корiнь вiдстанi до берега, тобто хвильове рiвняння має задану на кривiй особливiсть. Ця крива i визначає межу областi, в якiй вивчається задача. Основний результат роботи - ефективнi асимптотичнi формули для розв’язку зазначеноЁ задачi, включаючи окiл берега

Whitham method for Benjamin-Ono-Burgers equation and dispersive shocks in internal waves in deep fluid

The Whitham modulation equations for the parameters of a periodic solution are derived using the generalized Lagrangian approach for the case of damped Benjamin-Ono equation. The structure of the dispersive shock in internal wave in deep water is considered by this method.Comment: 8 pages, 4 figure

Structure and Crystallographic Texture Changes of Ferritic Martensitic Steel Resulting from Thermal Creep and Ageing Tests

Thermal ageing (650 and 700∘C during 1000, 7000 or 13300 h) and creep (700∘C, 50 MPa) tests of tubes made from ferritic-martensitic steels EK181 and ChS139 were carried out. With the aid of X-ray techniques the investigation of crystallographic texture and structure condition after tests was conducted. Thermal ageing provides substructure enhancement. With the increase of ageing time one can note the decrease of microhardness and X-ray peaks broadening, which indicates inner elastic microstress relaxation. It was revealed that changes of crystallographic texture in the rupture area of steel ChS139 tube after creep test is similar to those after uniaxial tensile test at room temeprature. This indicates the similarity of the mechanisms ofgrain reorientation for creep and tension. Recrystallization occurs in steel EK181 during creep test at temperature 700∘C leading to formation of recrystallization texture. This results in faster failure of steel EK181 (2486 h before rupture) in comparison with steel ChS139 (3426 h)

Hard loss of stability in Painlev\'e-2 equation

A special asymptotic solution of the Painlev\'e-2 equation with small parameter is studied. This solution has a critical point $t_*$ corresponding to a bifurcation phenomenon. When $t<t_*$ the constructed solution varies slowly and when $t>t_*$ the solution oscillates very fast. We investigate the transitional layer in detail and obtain a smooth asymptotic solution, using a sequence of scaling and matching procedures

Weakly-nonlocal Symplectic Structures, Whitham method, and weakly-nonlocal Symplectic Structures of Hydrodynamic Type

We consider the special type of the field-theoretical Symplectic structures called weakly nonlocal. The structures of this type are in particular very common for the integrable systems like KdV or NLS. We introduce here the special class of the weakly nonlocal Symplectic structures which we call the weakly nonlocal Symplectic structures of Hydrodynamic Type. We investigate then the connection of such structures with the Whitham averaging method and propose the procedure of "averaging" of the weakly nonlocal Symplectic structures. The averaging procedure gives the weakly nonlocal Symplectic Structure of Hydrodynamic Type for the corresponding Whitham system. The procedure gives also the "action variables" corresponding to the wave numbers of $m$-phase solutions of initial system which give the additional conservation laws for the Whitham system.Comment: 64 pages, Late

Approximate techniques for dispersive shock waves in nonlinear media

Many optical and other nonlinear media are governed by dispersive, or diffractive, wave equations, for which initial jump discontinuities are resolved into a dispersive shock wave. The dispersive shock wave smooths the initial discontinuity and is a modulated wavetrain consisting of solitary waves at its leading edge and linear waves at its trailing edge. For integrable equations the dispersive shock wave solution can be found using Whitham modulation theory. For nonlinear wave equations which are hyperbolic outside the dispersive shock region, the amplitudes of the solitary waves at the leading edge and the linear waves at the trailing edge of the dispersive shock can be determined. In this paper an approximate method is presented for calculating the amplitude of the lead solitary waves of a dispersive shock for general nonlinear wave equations, even if these equations are not hyperbolic in the dispersionless limit. The approximate method is validated using known dispersive shock solutions and then applied to calculate approximate dispersive shock solutions for equations governing nonlinear optical media, such as nematic liquid crystals, thermal glasses and colloids. These approximate solutions are compared with numerical results and excellent comparisons are obtained

Synthesis and Thermal Stability of Cubic ZnO in the Salt Nanocomposites

Cubic zinc oxide (rs-ZnO), metastable under normal conditions, was synthesized from the wurtzite modification (w-ZnO) at 7.7 GPa and ~800 K in the form of nanoparticles isolated in the NaCl matrix. The phase transition rs-ZnO \rightarrow w-ZnO in nanocrystalline zinc oxide under ambient pressure was experimentally studied for the first time by differential scanning calorimetry and high-temperature X-ray diffraction. It was shown that the transition occurs in the 370-430 K temperature range and its enthalpy at 400 K is -10.2 \pm 0.5 kJ mol-1.Comment: 12 pages, 4 figures, 1 tabl